Stone Coalgebras

@article{Kupke2003StoneC,
  title={Stone Coalgebras},
  author={Clemens Kupke and Alexander Kurz and Yde Venema},
  journal={Theor. Comput. Sci.},
  year={2003},
  volume={327},
  pages={109-134}
}
Algebraic Semantics for Coalgebraic Logics
Strongly Complete Logics for Coalgebras
TLDR
A general construction of a logic from an arbitrary set-functor is given and proven to be strongly complete under additional assumptions and it is argued that sifted colimit preserving functors are those functors that preserve universal algebraic structure.
Coalgebraic Geometric Logic: Basic Theory
Using the theory of coalgebra, we introduce a uniform framework for adding modalities to the language of propositional geometric logic. Models for this logic are based on coalgebras for an
Completeness for the coalgebraic cover modality
TLDR
A derivation system is introduced, and it is proved that it provides a sound and complete axiomatization for the collection of coalgebraically valid inequalities, and the Lindenbaum-Tarski algebra of the logic can be identified with the initial algebra for this functor.
Duality for powerset coalgebras
TLDR
Thomason duality is derive from Tarski duality, thus paralleling how Jónsson-TarskiDuality is derived from Stone duality.
A Coalgebraic Approach to Dualities for Neighborhood Frames
TLDR
An endofunctor is constructed on the category of complete and atomic Boolean algebras that is dual to the double powerset functor on Set to show that Thomason duality for neighborhood frames can be viewed as an algebra-coalgebra duality.
Coalgebraic Geometric Logic
TLDR
Using the theory of coalgebra, a uniform framework for adding modalities to the language of propositional geometric logic is introduced, and a method of lifting an endofunctor on $\mathbf{Set}$, accompanied by a collection of predicate liftings, to an end ofunctors on the category of topological spaces.
Enriched Stone-type dualities
Ultrafilter Extensions for Coalgebras
TLDR
The Jonsson-Tarski theorem giving a set-theoretic representation for each modal algebra and the bisimulation-somewhere-else theorem stating that two states of a coalgebra have the same (finitary modal) theory iff they are bisimilar (or behaviourally equivalent) in the ultrafilter extension of the coalgebra are generalised.
Presenting Functors by Operations and Equations
TLDR
A result is obtained that allows us to prove adequateness of logics uniformly for a large number of different types of transition systems and give some examples of its usefulness.
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